How do I snap one object into place with the opening of another object? (See picture)
If it's only about location and rotation around the Z axis, it should be as simple as:


Select the object you wanna align
Go to Editmode Tab
Select all A
Make sure you are in Vertex Select mode CtrlTab
Hold Shift and click a vertex twice, which you wanna use to align location(makes it the active vertex)
Enable Snapping ShiftTab
Make it snap to vertices CtrlShiftTab
Set Snap Target to Active
Grab your object G and snap it to the other model's corresponding vetex
Snap Cursor to Active ShiftS
Change Pivot Point to 3D Cursor .
Hold Shift and click a vertex twice, which you wanna use to lineup rotation(make it active)
Rotate the model R, constrain to Z axis and snap your vertex to the corresponding vertex of the other model(this can be really tricky, try to make the models a bit transparent to see what you are snapping to)


Not sure if the vertex in step 12 needs to have the same Z location as the first vertex... But even then, it should be possible to use any vertex with a bit more effort, to be able to constrain the right rotation axis in step 13 - using a custom transform orientation, which you would have to set up first...

If the models did not align originally, you might be able to snap to edge instead in step 13...